The Theory of Ceers Computes True Arithmetic

TL;DR

This paper demonstrates that various structures of computably enumerable equivalence relations (ceers) and their degrees are computationally as complex as true arithmetic, revealing deep logical equivalences.

Contribution

It establishes that the theory of ceers and related structures is 1-equivalent to true arithmetic, showing their interpretability of natural number arithmetic.

Findings

The theory of ceers is 1-equivalent to true arithmetic.

Dark and light ceers structures are also 1-equivalent to true arithmetic.

Interpretability of natural number arithmetic within these structures is demonstrated.

Abstract

We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of $\mathcal{I}$-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of $(\mathbb{N},+,\cdot)$.